Section 4.5 Binomial Theorem
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exeRciSeS 4.5
1. Expand the expression using the binomial theorem.
a. (a + b)
5
b. (x + y)
6
c. (a + 2)
5
d. (a − 4)
4
2. Expand the expression using the binomial theorem.
a. (2x + 3y)
3
b. (3x − 1)
5
c. (2p − 3q)
4
d. (3x + ½)
5
In Exercises 3–10, find the indicated term in the expansion.
3. The fourth term in (a + b)
10
4. The seventh term in (x − y)
12
5. The sixth term in (2x − 3)
9
6. The fifth term in (3a + 2b)
7
7. The last term in (x − 3y)
8
8. The last term in (ab + 3x)
6
9. The third term in (4x − 2y)
5
10. The fourth term in (3x − ½)
8
11. Use the binomial theorem (more than once) to expand (a + b + c)
3
.
12. Expand (1 + 0.1)
5
in order to compute (1.1)
5
.
13. What is the coefficient of x
3
y
4
in the expansion of (2x − y + 5)
8
?
14. What is the coefficient of x
5
y 
2
z 
2
in the expansion of (x + y + 2z)
9
?
15. Prove that
C(n + 2, r) = C(n, r) + 2C(n, r − 1) + C(n, r − 2) for 2 ≤ r ≤ n
(Hint: Use Pascal’s formula.)
16. Prove that
C(k, k) + C(k + 1, k) + c + C(n, k) = C(n + 1, k + 1) for 0 ≤ k ≤ n
(Hint: Use induction on n for a fixed, arbitrary k, as well as Pascal’s formula.)
S e c t i o n 4 . 5 review
tecHniQue
• Use the binomial theorem to expand a binomial.
• Use the binomial theorem to find a particular term
in the expansion of a binomial.
main iDeaS
• The binomial theorem provides a formula for expanding a binomial without multiplying it out.
• The coefficients of a binomial raised to a nonnegative integer power are combinations of n items as
laid out in row n of Pascal’s triangle.
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